$ A = \left[\begin{array}{rr}4 & 0 \\ 5 & 2 \\ 0 & -2\end{array}\right]$ $ F = \left[\begin{array}{rr}-2 & 4 \\ -2 & 1\end{array}\right]$ What is $ A F$ ?
Answer: Because $ A$ has dimensions $(3\times2)$ and $ F$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ A F = \left[\begin{array}{rr}{4} & {0} \\ {5} & {2} \\ \color{gray}{0} & \color{gray}{-2}\end{array}\right] \left[\begin{array}{rr}{-2} & \color{#DF0030}{4} \\ {-2} & \color{#DF0030}{1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rr}{4}\cdot{-2}+{0}\cdot{-2} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{4}\cdot{-2}+{0}\cdot{-2} & ? \\ {5}\cdot{-2}+{2}\cdot{-2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{4}\cdot{-2}+{0}\cdot{-2} & {4}\cdot\color{#DF0030}{4}+{0}\cdot\color{#DF0030}{1} \\ {5}\cdot{-2}+{2}\cdot{-2} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{4}\cdot{-2}+{0}\cdot{-2} & {4}\cdot\color{#DF0030}{4}+{0}\cdot\color{#DF0030}{1} \\ {5}\cdot{-2}+{2}\cdot{-2} & {5}\cdot\color{#DF0030}{4}+{2}\cdot\color{#DF0030}{1} \\ \color{gray}{0}\cdot{-2}+\color{gray}{-2}\cdot{-2} & \color{gray}{0}\cdot\color{#DF0030}{4}+\color{gray}{-2}\cdot\color{#DF0030}{1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-8 & 16 \\ -14 & 22 \\ 4 & -2\end{array}\right] $